We review the methodology to theoretically treat parity-time- ( PT -) symmetric, non-Hermitian quantum many-body systems. They are realized as open quantum systems with PT symmetry and couplings to the environment which are compatible. PT -symmetric non-Hermitian quantum systems show a variety of fascinating properties which single them out among generic open systems. The study of the latter has a long history in quantum theory. These studies are based on the Hermiticity of the combined system-reservoir setup and were developed by the atomic, molecular, and optical physics as well as the condensed matter physics communities. The interest of the mathematical physics community in PT -symmetric, non-Hermitian systems led to a new perspective and the development of the elegant mathematical formalisms of PT -symmetric and biorthogonal quantum mechanics, which do not make any reference to the environment. In the mathematical physics research, the focus is mainly on the remarkable spectral properties of the Hamiltonians and the characteristics of the corresponding single-particle eigenstates. Despite being non-Hermitian, the Hamiltonians can show parameter regimes, in which all eigenvalues are real. To investigate emergent quantum many-body phenomena in condensed matter physics and to make contact to experiments one, however, needs to study expectation values of observables and correlation functions. One furthermore, has to investigate statistical ensembles and not only eigenstates. The adoption of the concepts of PT -symmetric and biorthogonal quantum mechanics by parts of the condensed matter community led to a controversial status of the methodology. There is no consensus on fundamental issues, such as, what a proper observable is, how expectation values are supposed to be computed, and what adequate equilibrium statistical ensembles and their corresponding density matrices are. With the technological progress in engineering and controlling open quantum many-body systems it is high time to reconcile the Hermitian with the PT -symmetric and biorthogonal perspectives. We comprehensively review the different approaches, including the overreaching idea of pseudo-Hermiticity. To motivate the Hermitian perspective, which we propagate here, we mainly focus on the ancilla approach. It allows to embed a non-Hermitian system into a larger, Hermitian one. In contrast to other techniques, e.g. master equations, it does not rely on any approximations. We discuss the peculiarities of PT -symmetric and biorthogonal quantum mechanics. In these, what is considered to be an observable depends on the Hamiltonian or the selected (biorthonormal) basis. Crucially in addition, what is denoted as an ‘expectation value’ lacks a direct probabilistic interpretation, and what is viewed as the canonical density matrix is non-stationary and non-Hermitian. Furthermore, the non-unitarity of the time evolution is hidden within the formalism. We pick up several model Hamiltonians, which so far were either investigated from the Hermitian perspective or from the PT -symmetric and biorthogonal one, and study them within the respective alternative framework. This includes a simple two-level, single-particle problem but also a many-body lattice model showing quantum critical behavior. Comparing the outcome of the two types of computations shows that the Hermitian approach, which, admittedly, is in parts clumsy, always leads to results which are physically sensible. In the rare cases, in which a comparison to experimental data is possible, they furthermore agree to these. In contrast, the mathematically elegant PT -symmetric and biorthogonal approaches lead to results which, are partly difficult to interpret physically. We thus conclude that the Hermitian methodology should be employed. However, to fully appreciate the physics of PT -symmetric, non-Hermitian quantum many-body systems, it is also important to be aware of the main concepts of PT -symmetric and biorthogonal quantum mechanics. Our conclusion has far reaching consequences for the application of Green function methods, functional integrals, and generating functionals, which are at the heart of a large number of many-body methods. They cannot be transferred in their established forms to treat PT -symmetric, non-Hermitian quantum systems. It can be considered as an irony of fate that these methods are available only within the mathematical formalisms of PT -symmetric and biorthogonal quantum mechanics.

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PT - 对称、非厄米量子多体物理——方法论视角


我们回顾了从理论上处理宇称时间（PT）对称、非厄米量子多体系统的方法。它们被实现为具有 PT 对称性和与环境兼容的耦合的开放量子系统。 PT 对称非厄米量子系统表现出各种令人着迷的特性，这些特性使它们在通用开放系统中脱颖而出。后者的研究在量子理论中有着悠久的历史。这些研究基于组合系统-储层设置的厄米性，并由原子、分子和光学物理学以及凝聚态物理学界开发。数学物理界对 PT 对称、非厄米系统的兴趣带来了新的视角，并发展了 PT 对称和双正交量子力学的优雅数学形式，这些形式不涉及任何环境。在数学物理研究中，焦点主要集中在哈密顿量显着的光谱性质以及相应的单粒子本征态的特征上。尽管是非厄米特式，哈密顿量可以显示参数状态，其中所有特征值都是实数。然而，为了研究凝聚态物理中涌现的量子多体现象并与实验建立联系，需要研究可观测量和相关函数的期望值。此外，必须研究统计系综而不仅仅是本征态。凝聚态物质界的部分人采用 PT 对称和双正交量子力学的概念导致了该方法论的争议。 在基本问题上尚未达成共识，例如什么是适当的可观测量、如何计算期望值以及什么是充分的平衡统计集合及其相应的密度矩阵。随着工程和控制开放量子多体系统的技术进步，现在是协调厄米特与 PT 对称和双正交观点的时候了。我们全面回顾了不同的方法，包括伪隐秘性的过度思想。为了激发我们在这里传播的埃尔米特观点，我们主要关注辅助方法。它允许将非厄米系统嵌入到更大的厄米系统中。与其他技术相比，例如主方程，它不依赖于任何近似值。我们讨论 PT 对称和双正交量子力学的特性。在这些中，什么被认为是可观察的取决于哈密顿量或所选的（双正交）基础。此外，至关重要的是，所谓的“期望值”缺乏直接的概率解释，而所谓的规范密度矩阵是非平稳和非厄米式的。此外，时间演化的非统一性隐藏在形式主义之中。我们选择了几种模型哈密顿量，迄今为止这些模型要么是从厄米特角度进行研究，要么是从 PT 对称和双正交角度进行研究，并在各自的替代框架内研究它们。这包括一个简单的两级单粒子问题，还包括一个显示量子临界行为的多体晶格模型。 比较两种类型的计算结果表明，厄米特方法虽然有些笨拙，但总是能得出物理上合理的结果。在极少数情况下，可以与实验数据进行比较，他们还同意这些。相比之下，数学上优雅的 PT 对称和双正交方法产生的结果在某种程度上难以物理解释。因此我们得出的结论是应该采用埃尔米特方法。然而，为了充分理解 PT 对称、非厄米量子多体系统的物理原理，了解 PT 对称和双正交量子力学的主要概念也很重要。我们的结论对格林函数方法、泛函积分和生成泛函的应用产生了深远的影响，这些是大量多体方法的核心。它们不能以其既定形式转移来处理 PT 对称、非厄米量子系统。这些方法只能在 PT 对称和双正交量子力学的数学形式中使用，这可以被认为是命运的讽刺。